176 VIII DIFFERENTIAL CALCULUS

,...»w, where the cpt are scalar functions defined in E, and by
(8.1 .5) /is continuously differentiate if and only if each of the <pt is con-
tinuously differentiate; again, by case 1, q>i is continuously differentiable
if and only if each of the partial derivatives Dj<pt (which is now a scalar
function) exists and is continuous. Furthermore, the (total) derivative of
/is the linear mapping

with

in other words,/', which is a linear mapping of R" into Rm (resp. of Cn into
Cm), corresponds to the matrix (Dj<pfai9..., an)), which is called the jacobian
matrix of / (or of <pi9..., <pm) at (al9..., aw). When m~n, the determinant
of the jacobian (square) matrix of/is called the jacobian off (or of <p1?..,, <prt).
Theorem (8.9.2) specializes to

(8.10.1) Let (pj (1 < j ^ m) be m scalar functions, continuously differentiable
in an open subset A 0/R" (resp. C1); let \l/t (1 < / </?) 6e /? scalar functions,
continuously differentiable in an open subset B 0/ Rm (resp. Cm) containing
the image of A 6y (^, . . . , (pm); then if Ofa) = ^(^(x), . . . , <pm(x)) /or
x e A and 1 < / < p, we have the relation

between the jacobian matrices; in particular, when m = n =p, we have the
relation

detO^fli) = det(D,.^) det(Dfc^-)
between thejacobians.

We mention here the usual notations fifa, . . . , £„), TJ-

for D^/Ki, ..., fj, which unfortunately lead to hopeless confusion when
substitutions are made (what does /;(y, x) or /^fox) mean?); the
jacobian det(D^£(^, . . . , Q) is also written D(<p1? . . . , ^)/D(^, . . . , £„) or

11. DERIVATIVE OF AN INTEGRAL DEPENDING ON A PARAMETER

(8.11.1) Let I = [a, ft] c R ie a compact interval, E, F real Banach spaces,
f a continuous mapping of I x A into F (A open subset of E). Then
g(z) = j /(£, z) d£ is continuous in A.

Jawith r = (x2 +y2)1/2> and/(0,0) = 0. Show that Di/and D2/ exist at every point